r = a + μm, a and m are constant vectors and μ is a parameter (a numerical value can vary)

Notice that this is very similar to the equation of the line in 2D space, y = mx + c

① In 2D space, c is the y-intercept, a fixed point.

Likewise, in 3D space, a is the position vector of a fixed (therefore constant) point.


② In 2D space, m is the gradient of the line, a fixed/constant value. This gradient gives the slope/direction of the line.

Likewise, in 3D space, m is a direction vector that is fixed and constant. It provides the direction of a line and is parallel to it.


③ In 2D space, y and x are coordinates of a point that lie on the line. They are variables. x is the independent variable and y is the dependent variable.

y is determined by x.

Likewise, in 3D space, r is a vector that is determined by the independent variable (parameter) μ.


Comparing side by side,

y = mx + c
r = mμ + a


So geometrically,

r = a + μm represents a line in 3D space, whereby :

- r is the position vector of any point on the line.

- a is the position vector of a fixed point that the line passes through.

- m is the direction vector that is parallel to the line

- μ is a parameter that is variable, which determines r , μ ∈ R (belongs to the set of real values)